## Essential University Physics: Volume 1 (3rd Edition)

We know that $a_r=\frac{v^2}{r}$ Differentiating both sides w.r.t 't' , we obtain: $\frac{da_r}{dt}=\frac{d}{dt}(\frac{v^2}{r})$ $\implies \frac{da_r}{dt}=\frac{r\frac{dv^2}{dt}-v^2\frac{dr}{dt}}{r^2}$ $\implies \frac{da_r}{dt}=\frac{2vr\frac{dv^2}{dt}-v^2\frac{dr}{dt}}{r^2}$ $\implies \frac{da_r}{dt}=\frac{2vrat-v^2\frac{dr}{dt}}{r^2}$ $\implies \frac{da_r}{dt}=\frac{2vat}{r}-\frac{v^2}{r^2}\frac{dr}{dt}$